At this point, I'm thinking there are two types of emergency managers. One set has decided to implement the use of social media into their emergency management functions and the second set have said, "No thank you, and since no one is demanding it, I can coast and not do anything." This latter group will get squeezed by citizens and the media when something does happen.
With the above in mind, see this infographic, Social Media the New Face of Disaster Response. The 2010 Haiti earthquake was the first widespread use of Twitter during a disaster. That event is now six years in our rearview mirror, so don't think you are being cutting edge if you have established a Facebook page that is basically static.
The other thing that can be disturbing to both sets of emergency managers I outlined above are the public's expectations, also outlined in the infographic for response agencies to receive requests for assistance via social media from citizens. Just saying "We don't do that" will not ring well when others in the nation are grappling with how to accomplish this.
These are challenging times and they require people and agencies to go beyond budgets and staffing levels to find solutions.